EVEN AND ODD FUNCTIONS

EVEN FUNCTIONS

for even functions:
when inputs are opposites, the corresponding outputs are the same

Recall that $\,x\,$ and $\,-x\,$ are opposites.
When $\,x\,$ is the input, $\,f(x)\,$ is the output.
When $\,-x\,$ is the input, $\,f(-x)\,$ is the output.
For even functions, these two outputs must be equal.

Note: A function $\,f\,$ is even if and only if the graph of $\,f\,$ is
symmetric about the $\,y\,$-axis.

Note: $\,y = x^2\,$, $\,y = x^4\,$, $\,y = x^6\,$, $\,y = x^8\,$, etc., are all even functions.
Thus, power functions with even powers are even functions!
So, the name ‘even’ seems reasonable.

There are lots of other even functions.
For example, $\,y = |x|\,$ and $\,y = \cos(x)\,$ are even functions.

Is $\,g(t) = t^2 - t^4\,$ an even function?
You must investigate $\,g(-t)\,$, and compare it with $\,g(t)\,$, as follows:

ODD FUNCTIONS

for odd functions:
when inputs are opposites, the corresponding outputs are opposites

Recall that $\,x\,$ and $\,-x\,$ are opposites.
When $\,x\,$ is the input, $\,f(x)\,$ is the output.
When $\,-x\,$ is the input, $\,f(-x)\,$ is the output.
For odd functions, these two outputs must be opposites.

Observe that ‘$\,f(x) = -f(-x)\,$’ is equivalent to
‘$\,f(-x) = -f(x)\,$’ (multiply both sides by $\,-1\,$ and switch sides).
Sometimes, the definition is given using this equivalent characterization.